# METHODOLOGY AND METHODS FOR CREATING MATHEMATICS PROBLEMS FOR SECONDARY SHOOL

**Journal**: Bulletin of Cherkasy University. Pedagogical Sciences (Vol.2017, No. 16)

**Publication Date**: 2017-12-13

**Authors** : MILLOUSHEVA-BOYKINA D.; MILLOUSHEV V.;

**Page** : 103-114

**Keywords** : problem; methodology; methods for creating problems; analogy; generalization; specialization; concretization;

### Abstract

Introduction. It is well known that the process of solving heuristic mathematical problems of non-algorithmic type is a creative activity in which not only mathematical knowledge is applied but also intuition and imagination. Of course, creativity is more prominent in the process of creating problems, since this activity requires not only deep knowledge of mathematical objects, but also a deeper understanding of the structure of the problems and revealing the mathematical abilities of the subject. Purpose. The aim of the article is to present certain basic problems and methodology of applying the methods of analogy, generalization, specialization and concretization in creating mathematical problems for the secondary school. The application of these methods is illustrated with series of problems created from a given basic one, revealing the relations between the created problems considered as a system. Methods. Theoretical analyses of the activity «creating» problems and experimental application of the methods analogy, generalization, specialization and concretization in practice. Results. The actuality of carrying out the activity of creating problems for the school course of mathematics is motivated in the article. A certain methodology for creating problems by the methods of analogy, generalization, specialization, concretization is presented. In order to realize this methodology the learner first have to make an analysis of the formulation of a certain problem from the school course of mathematics, which we call «a basic problem». Sometimes the formulation of the basic problem is not suitable for making a generalization. In such cases, on the base of the analysis that we have made, we make a new formulation of the problem which is better for generalization. Usually in each problem there are given some restrictions about some of the objects in it or there are constants. In such cases generalization can be made in the following ways: a) by removing a given restriction; b) by replacing of a constant with a parameter; c) combining a) and b). For convenience in the paper we indicate the basic problems in the following way: Problem 1, Problem2, …, while the problems that are created from the basic Problem 1 using analogy – with Problem 1а1, Problem 1а2, …; and the problems that are created by generalization by removing a certain restriction or by replacing of a constant with a parameter we indicated as: Problem 1о1, Problem 1о2, … If we remove another restriction, the certain generalized problems received from Problem 1а1, we indicate with: Problem 1а1о1, Problem 1а1о2 and so on. Schematic models of the relations between the created problems, which are considered as a system, are presented. Originality. Teaching students to solve problems from the school course in mathematics is a popular in the curricular. But the question about teaching students – future teachers in mathematics in creating problems is not developed enough. That's why we have created a special course for our university students named «Methods and methodology for creating problems» which gives students practical skills for creating and solving problems. This leads to improvement the quality of their professional training as future teachers in mathematics. Conclusion. As a conclusion we shall point that the presented models serve to illustrate the processes of applying the methods of analogy, generalization, specialization, concretization for creating problems and to reveal the relations between the basic problem and the created ones as well as between the created problems themselves. On the other hand this process gives the opportunity to put different problems in front of the students – such as to find out new problems, which they have to formulate and justify. Thus, creative mathematical activity on the part of the students takes place.

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Last modified: 2018-06-13 19:15:33